# Linear or Nonlinear Modeling for ENSO Dynamics?

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## Abstract

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## 1. Non-Gaussian-Nonlinear Features of the ENSO Statistics

## 2. The Dynamics of the Components of the ENSO: The Recharge Oscillator Model

## 3. Is a Possible Internal Nonlinearity Relevant?

## 4. The Multiplicative Nature of the Forcing

## 5. The Fokker–Planck Equation Guiding the Statistics of the ROM

## 6. Inference of the Statistical Features of the ROM from Observations

#### 6.1. Linear Equation of Motion for the First Two Moments of the ROM and the White Noise Approximation

- analyzing only the first and second moments/cumulants/cross-correlation functions of the observations data, we cannot identify/detect the (possible) nonlinearity due to the interaction with the atmosphere, in the ENSO system;
- comparing the first and second moments/cumulants/cross-correlation functions, we obtain from the FPE of Equation (18) with observations, it would be really hard to determine how small the time scale $\tau $ of decaying of the correlation function of the effective noise perturbing the ROM is. On the other hand, as has been shown in [17], also for $\tau \to 0$, the FPE of Equation (18) well accounts for the non-Gaussianity of the ENSO statistics. Thus, for the sake of simplicity, from now on, we shall use the FPE of Equation (17), which is the white noise limit of Equation (18), even if we are aware of the fact that the time scale separation between the dynamics of the averaged ROM and that of the fast atmosphere is not so “extreme”.

#### 6.2. The Covariance Matrix of the ROM and the Comparison with the ENSO Data

#### 6.3. Signatures of a Nonlinear Perturbation: Skewed Stationary PDF and High Frequency of Strong Events

#### 6.4. Inferring the FPE Coefficients from Data

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Validation of the Linear Relationship between the Wind Stress and T_{E}

- (1)
- From Figure A2, we clearly see that the time behaviors of ${\tau}_{s,slow}$ and ${T}_{E,slow}$ are similar (of course, some lag remains between the wind stress, which is a a forcing term, and the ocean reaction);
- (2)
- From Figure A3, we see that the plot of the cross-correlation between ${\tau}_{s,slow}$ and ${T}_{E,slow}$ vs. time (in moths) is very similar to the plot of the autocorrelation of ${T}_{E,slow}$ (notice the little time offset between these two plots, indicating the fact that the one-year time average has not completely hidden the cause-effect relationship between ${\tau}_{s,slow}$ and ${T}_{E,slow}$). Moreover, from the insert of the same figure, we see that the average relationship between ${\tau}_{s,slow}$ and ${T}_{E,slow}$ is mainly linear, with a small dispersion of the data around the linear fit (quantities are normalized by the standard deviation of the quantities from observations):$${\tau}_{s}\equiv {\tau}_{s,slow}+{\tau}_{s,fast}=k{T}_{E}+{\tau}_{s,fast}.$$

**Figure A1.**5N–5S averaged zonal wind for different longitudes. Data from NOAA [61].

## Appendix B. Very Short Review of the Projection Approach Applied to the ROM

## Appendix C. The Ansatz and the Stationary PDF

**Figure A2.**Plot of the 13-month averaged Niño 3 data (blues solid line) and of the 13-month averaged wind stress anomaly (dashed red line). Quantities are normalized by their respective standard deviations. The Niño 3 data are from [50], while the wind stress is proportional to the square of the wind velocity, where the wind data are from [61]. The wind stress has been averaged on the equatorial strip defined by (5${}^{\circ}$ S–5${}^{\circ}$ N) × (120${}^{\circ}$ W–180${}^{\circ}$ W).

**Figure A3.**Blue solid line: normalized auto-correlation of the one-year averaged Niño 3 index: ${\langle {T}_{E,slow}\left(0\right){T}_{E,slow}\left(t\right)\rangle}_{s}/{\langle {T}_{E,slow}\left(0\right){T}_{E,slow}\left(0\right)\rangle}_{s}$. Red-dashed line: cross-correlation between the ${\tau}_{s,slow}$ and ${T}_{E,slow}$, normalized with the initial value: ${\langle {\tau}_{s,slow}\left(0\right){T}_{E,slow}\left(t\right)\rangle}_{s}/{\langle {\tau}_{s,slow}\left(0\right){T}_{E,slow}\left(0\right)\rangle}_{s}$. We can see that they are very similar to each other. Insert: ${\tau}_{s,slow}$ vs. ${T}_{E,slow}$ (red dots). Data from NOAA [50] (Niño 3) and [61] (wind).

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**Figure 1.**Histogram of the frequency for the three-month average Niño 3 data from NOAA [50]. The dashed line is the best fit of the stationary PDF found in [17] (strongly skewed and with a heavy tail), and solid blue line is a normal distribution with the same average and variance of the observation data.

**Figure 2.**The values of the coefficients ${A}_{0}$, ${A}_{4}$ and ${A}_{3}$ vs. $\tau $ for $\phi \left(t\right)=exp(-t/\tau )$ for $1/\lambda =12$ months. In the shown range 0 months $\le \tau \le 3$ months, ${A}_{4}\sim {A}_{0}$ and ${A}_{3}\sim 0$. We recall that ${A}_{2}={A}_{4}+{A}_{0}$ and that ${A}_{1}={A}_{3}$. Thus, $A\sim {A}_{0}{(1+\beta T)}^{2}$, which must be compared to $A=D{(1+\beta T)}^{2}$, which holds true in the limit of a very large time scale separation between the dynamics of the ROM and that of the booster (see the text for details).

**Figure 3.**Evolution of the variance of T (left), h (center) and of the cross-correlation between T and h, obtained from Equation (20b) with the ${A}_{i}$ and ${B}_{i}$ coefficients given by Equations (A7)–(A14), in the case where $1/\lambda =12$ month and $\phi \left(t\right)=exp(-t/\tau )$, for different values of $\tau $. Solid line: $\tau =6$ months; dashed line: $\tau =3$ months; dotted line: $\tau =1$ month. Curves for different $\tau $ values are very similar.

**Figure 4.**Correlation matrix ${\langle x\left(t\right)y\left(0\right)\rangle}_{s}/{\langle {y}^{2}\rangle}_{s}$, where “x” and “y” can be either h or T. Solid line: from the NOAA data (the h values are taken from [76] and refer to the anomalies of the Volume of Warm Water (WWV) in the basin 120${}^{\circ}$ E–80${}^{\circ}$ W, 5${}^{\circ}$ S–5${}^{\circ}$ N, in the time range from January 1982–December 2017). Dashed line, from the Fokker–Planck Equation (FPE) of the Recharge Oscillator Model (ROM), i.e., from Equations (27) and (28) in the case where $\omega =2\pi $/48 month${}^{-1}$ and $\lambda =1/12$ month${}^{-1}$. Dotted line: a fit with the same functions of Equations (27) and (28).

**Figure 5.**Solid blue line: periodogram of the Niño 3 data from NOAA evaluated averaging over non-overlapping partitions of length 300 months. Dashed black line: theoretical power spectrum of an ROM with $\omega =2\pi $/48 month${}^{-1}$, $\lambda =1/10$ month${}^{-1}$ and additive white noise perturbation.

**Figure 6.**The month averaged Niño 3 data from January 1949–February 2016. Data are from the Tokyo Climate Center, WMO Regional Climate Centers (RCCs) [78]. In green, we highlight a segment of the time series to illustrate the First Passage Time (FPT) $\delta t({T}_{i},{T}_{tg})$ for a given target temperature anomaly ${T}_{tg}$ (here, ${T}_{tg}=1.5$), starting from an initial neutral temperature ${T}_{i}$ ($-0.5\le {T}_{i}\le 0.5$). See also the text and Figure 7 for details about the FPT.

**Figure 7.**The first passage time $\delta t({T}_{i},{T}_{tg})$ for a target temperature anomaly of ${T}_{tg}=1.5$ is defined as the first time the fluctuating temperature $T\left(t\right)$ crosses the threshold ${T}_{tg}$ ($1.5$ in this case), starting from an initial temperature ${T}_{i}$ (that we will choose in the range $-0.5\le {T}_{i}\le 0.5$, corresponding to neutral ENSO conditions).

**Figure 8.**The mean FPT for different values of the $\lambda $ parameter, vs. the target temperature, obtained using Equation (32), compared with observations (circles). The values of $\beta $ and $\mu $ have been fixed by fitting the stationary PDF of Equations (29)–(31) to Niño 3 from [50]: $\beta =0.2$ and $\mu =32.7$ (see the text for details). Dashed line: $\lambda =1/6$ month${}^{-1}$, dotted-dashed line $\lambda =1/8$ month${}^{-1}$, solid line $\lambda =1/12$ month${}^{-1}$. In the inset, a zoom of the same graphs, where the gray background emphasizes the range of 2–7 years and $1.0\le T\le 1.6$, corresponding to the typical recurring time interval for intermediate El Niño events. Notice how the curve obtained with $\lambda =1/12$ month${}^{-1}$ falls better than the others in this zone.

**Figure 9.**Semi-log plot of the average FPT as a function of the target temperature ${T}_{tg}$, for $\lambda =1/12$ month${}^{-1}$ and ${\sigma}^{2}=D/\lambda =0.8$ (see the text for details). Thick and thin solid lines show analytical solutions from Equation (32) with $\beta =0.2$ and $\beta =0.3$, respectively; while the thick dashed line is for the case $\beta =0$, corresponding to the pure additive forcing of the ROM (in this case, the stationary PDF is the Gaussian). Circles: the average FPT from Niño 3 data from NOAA [50]. Crosses: average FPT from numerical simulations of the Îto SDE corresponding to the FPE of Equation (17) [79]: $\mathrm{d}h=-\omega T\mathrm{d}t$; $\mathrm{d}T=(-\omega h+D\beta )\mathrm{d}t-(\lambda -D{\beta}^{2})T\mathrm{d}t+\sqrt{D}(1+\beta T),\mathrm{d}W$, where W is a Wiener process, $\beta =0.2$, and $\omega =2\pi /48$ month${}^{-1}$.

**Figure 10.**Dots: the expectation value $\mathbb{E}[T(t+\delta t)-T(t)]$ where $\delta t=1$ month, of the Niño 3 data from NOAA [50] compared with the drift coefficient ${\langle {G}_{T}(h,T)\rangle}_{T,s}\simeq -\lambda T$ (see Equation (37)), in the cases where the coefficient $\lambda $ is obtained as the the best fit to the data (dashed green line), from which $\lambda =1/13.7$ month${}^{-1}$, close to the case where $\lambda =1/12$ month${}^{-1}$ (solid red line) that corresponds to the value obtained fitting the correlation functions and the average timing of the ENSO events (see the text for details). Gray dashed line: number of observed events for each value of T, scaled by a factor 1/76 in the vertical axes. For large absolute values of T, the spreading of the data around the linear fit increases because the number of registered ENSO events decreases (e.g., just one for $T=-2$).

**Figure 11.**Dots: the expectation value $\mathbb{E}[{(T(t+\delta t)-T\left(t\right))}^{2}]/2$ where $\delta t=1$ month, of the Niño 3 data from NOAA [50]. Solid red line: diffusion term $A\left(T\right)=D{(1+\beta T)}^{2}$ (see Equation (36)), where $\beta =0.2$ and $D={\sigma}^{2}\lambda =0.07$ month${}^{-1}$ (see the text for details). Gray dashed line: the number of observed events for each value of T, scaled by a factor 1/160 in the vertical axes. The points for extreme events are not that close to the red line, but in these cases, the statistics is also really poor. Green dotted line, function $D{(1+\beta T)}^{2}+{D}_{1}$, with the following figures obtained from the fit to the data with fit parameters $\beta $ and D (${D}_{1}={\sigma}^{2}\lambda -D$ to ensure a fixed variance of the PDF): $\beta =0.9$, $D=0.011$ month${}^{-1}$ and ${D}_{1}={\sigma}^{2}\lambda -D=0.07$ month${}^{-1}-0.011$ month${}^{-1}=0.059$ month${}^{-1}$.

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**MDPI and ACS Style**

Bianucci, M.; Capotondi, A.; Mannella, R.; Merlino, S.
Linear or Nonlinear Modeling for ENSO Dynamics? *Atmosphere* **2018**, *9*, 435.
https://doi.org/10.3390/atmos9110435

**AMA Style**

Bianucci M, Capotondi A, Mannella R, Merlino S.
Linear or Nonlinear Modeling for ENSO Dynamics? *Atmosphere*. 2018; 9(11):435.
https://doi.org/10.3390/atmos9110435

**Chicago/Turabian Style**

Bianucci, Marco, Antonietta Capotondi, Riccardo Mannella, and Silvia Merlino.
2018. "Linear or Nonlinear Modeling for ENSO Dynamics?" *Atmosphere* 9, no. 11: 435.
https://doi.org/10.3390/atmos9110435